detp_neg_one_one — Mathlib

English

A particular case of det_fromBlocks₂₂ when the bottom-right block is the identity; det equals det(A − B C).

Русский

Особый случай det_fromBlocks₂₂, когда D = I, даёт det(fromBlocks(A,B,C,I)) = det(A − B C).

LaTeX

$$$\\det\\mathrm{fromBlocks}(A,B,C,I) = \\det\\bigl(A - B C\\bigr)$$$

Lean4

@[simp]
theorem detp_neg_one_one : detp (-1) (1 : Matrix n n R) = 0 :=
  by
  rw [detp, sum_eq_zero]
  intro σ hσ
  have hσ1 : σ ≠ 1 := by
    contrapose! hσ
    rw [hσ, mem_ofSign, sign_one]
    decide
  obtain ⟨i, hi⟩ := not_forall.mp (mt Perm.ext_iff.mpr hσ1)
  exact prod_eq_zero (mem_univ i) (one_apply_ne' hi)

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